Do $\binom{n}{i}$ and $\binom{n}{j}$ always share a prime factor $p \ge i$? (Erdős #699)

Is it true that for every $1\leq i<j\leq n/2$ there exists some prime $p\geq i$ such that $$p\mid \gcd\left(\binom{n}{i}, \binom{n}{j}\right)?$$ In words: any two binomial coefficients from the first half of row $n$ of Pascal's triangle must share a common prime factor at least as large as the smaller of the two indices. A single triple $(n,i,j)$ with $1\le i<j\le n/2$ for which every common prime factor of $\binom{n}{i}$ and $\binom{n}{j}$ is smaller than $i$ would disprove the conjecture. (Note that whether a prime $p$ divides $\binom{n}{k}$ is decided by Kummer's theorem: $p\mid\binom{n}{k}$ if and only if adding $k$ and $n-k$ in base $p$ produces at least one carry.)

Acceptance. FULLY RESOLVES: (a) a counterexample — an explicit triple $(n,i,j)$ with $1\le i<j\le n/2$, together with the complete list of common prime factors of $\binom{n}{i}$ and $\binom{n}{j}$ (certified, e.g., via Kummer-carry verification code that a reviewer can run), showing every common prime factor is $<i$; or (b) a proof that the required prime $p\ge i$ always exists — machine-checkable (Lean, building on the existing formal statement) preferred, else a complete written proof. ADVANCES: (a) exhaustive verification of the conjecture for all $n$ up to a stated record bound N, with the search program and a reproducible exhaustiveness certificate; (b) new counterexamples to the STRONG form ($p>i$) with $i\ge 4$ — only one is currently known — or a proof that the known list of strong-form exceptions is complete in some verified range; (c) a proof of the conjecture for a restricted regime (e.g., fixed small $i$, or $j$ in a bounded window of $i$). Deliver the witness triple with factorization and checking code, or the search code plus the attained verification bound, or the proof file.